A FRACTIONAL ORDER SEIR MODEL WITH DENSITY DEPENDENT DEATH RATE

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1 Hacettepe Journal of Mathematics and Statistics Volume 4(2) (211), A FRACTIONAL ORDER SEIR MODEL WITH DENSITY DEPENDENT DEATH RATE Elif Demirci, Arzu Unal and Nuri Özalp Received 21:6 :21 : Accepted 2 :1 :21 Abstract In this paper, we introduce a fractional order SEIR epidemic model with vertical transmission, where the death rate of the population is density dependent, i.e., dependent on the population size. It is also assumed that there exists an infection related death rate. We show the existence of nonnegative solutions of the model, and also give a detailed stability analysis of disease free and positive fixed points. A numerical example is also presented. Keywords: Fractional derivative, Initial value problem, SEIR model, Stability, Numerical solution. 21 AMS Classification: Primary: 5 C 38, 15 A 15. Secondary: 5 A 15, 15 A Introduction Communicated by Bulent Karasözen Mathematical modeling in epidemiology provides new aspects in understanding the spread of diseases, and it suggests control strategies [3]. One of the early models in epidemiology was introduced in 1927 [9] to predict the spreading behaviour of a disease. Since then, many epidemic models have been derived [8]. In [7] a detailed analysis for integer order SEIR models with vertical transmission within a constant population can be found. There are also several papers [12, 13] about epidemic models within a nonconstant population, which is more realistic. Although a large number of works has been done on modeling the dynamics of epidemiological diseases, it has been restricted to integer order (delay) differential equations. In recent years, it has turned out that many phenomena in different fields can be described very successfully by the models using fractional order differential equations [1,2,4,6]. Ankara University, Faculty of Sciences, Department of Mathematics, 61 Besevler, Ankara, Turkey. (E. Demirci) edemirci@science.ankara.edu.tr (A. Unal) aogun@science.ankara.edu.tr (N. Özalp) nozalp@science.ankara.edu.tr Corresponding Author.

2 288 E. Demirci, A. Unal, N. Özalp In this paper, we first introduce a fractional order SEIR model with vertical transmission in a population with density dependent death rate. This gives a logistic growth in the population for a specific, more realistic death rate. We show the nonnegativeness of the solutions and also give a detailed stability analysis. Finally, numerical simulations are presented to illustrate the obtained results. We begin by giving the definitions of fractional order integrals and derivatives [15]. For fractional order differentiation, we will use Caputo s definition, due to its convenience for initial conditions of the differential equations Definition. The fractional integral of order α > for a function f : R + R is defined by I α f (t) = 1 Γ (α) t (t τ) α 1 f (τ) dτ, and the Caputo fractional derivative of order α (n 1, n) of f(t) is defined by D α f (t) = I n α D n f (t), with n 1 being the integer part of α and D = d/dt. Here and elsewhere, Γ denotes the Gamma function. Note that under natural conditions on the function f (t), for α n the Caputo derivative becomes the conventional derivative [15]. 2. Model derivation Many infectious diseases in nature have both horizontal and vertical transmission routes. These include such human diseases as Rubella, Herpes Simplex, Hepatitis B, Chagas, and the HIV/AIDS. Horizontal transmission of diseases among humans and animals, occurs through physical contact with hosts or through disease vectors like mosquitos, flies, etc. Vertical transmission is the transmission of an infection from parents to child during the perinatal period. The model that we study in this paper is a fractional order SEIR epidemic model with vertical transmission. The total host population N(t) is partitioned into four compartments which are susceptible, exposed, infectious and recovered, with sizes denoted by S(t), E(t), I(t) and R(t), respectively. Let b denote the natural birth rate of the population. The horizontal transmission of the disease is assumed to take place with direct contact between infectious and susceptible hosts with a transmission rate r, which means is the number of infections caused by all infected individuals per unit of time. For the vertical transmission of the disease, we assume that the offsprings of exposed and infectious classes are born into the exposed class with probabilities of p and q, respectively. We also assume that the natural death rate d(n) depends on the size of the population. For convenience, d is assumed to be a continuous and non decreasing function on R +, which does not contradict with biological phenomena. Also we assume that there exists a positive constant K which represents the carrying capacity of the population, and such that d (K) = b. We shall note that if d(n) is a linear function then the population has a logistic growth. r SI N These assumptions lead to the following system of differential equations of order α, with β, γ > being the rate that exposed individuals become infectious and the recovery

3 Fractional Order SEIR Model 289 rate, respectively, and θ is the infection related death rate: (2.1) (2.2) D α S = bn pbe qbi r SI N d(n)s, D α E = pbe + qbi + r SI βe d(n)e, N D α I = βe θi γi d(n)i, D α R = γi d(n)r, S () = S, E () = E, I () = I, R () = R, where < α 1, N = S + E + I + R, (S, E,I, R) R 4 +. The reason for considering a fractional order system instead of its integer order counterpart is that fractional order differential equations are generalizations of integer order differential equations. Also, using fractional order differential equations can help us to reduce the errors arising from the neglected parameters in modeling real life phenomena. We should note that the system (2.1) can be reduced to an integer order system by setting α = 1. Adding up the equations given in (2.1), we have (2.3) D α N = N(b d(n)) θi, which means the population size is not constant. 3. Non-negative solutions Let R 4 + = {X R 4 : X } and X (t) = (S (t),e (t),i (t), N (t)) T. For the proof of the theorem about non-negative solutions we shall need the following Lemma [14]: 3.1. Lemma. (Generalized Mean Value Theorem) Let f (x) C[a, b] and D α f (x) C(a,b] for < α 1. Then we have f (x) = f (a) + 1 Γ(α) Dα f (ξ) (x a) α with ξ x, x (a, b] Remark. Suppose f (x) C[, b] and D α f (x) C(, b] for < α 1. It is clear from the Lemma 3.1 that if D α f (x), x (, b), then the function f is nondecreasing, and if D α f (x), x (, b), then the function f is nonincreasing for all x [, b] Theorem. There is a unique solution for the initial value problem given by (2.1)- (2.2), and the solution remains in R 4 +. Proof. The existence and uniqueness of the solution of (2.1)-(2.2) in (, ) can be obtained from [1, Theorem 3.1 and Remark 3.2]. We need to show that the domain R 4 + is positively invariant. Since D α S S= = bn pbe qbi, D α E E= = qbi + r SI N, D α I I= = βe, D α R R= = γi, on each hyperplane bounding the nonnegative orthant, the vector field points into R 4 +.

4 29 E. Demirci, A. Unal, N. Özalp It is clear that N(t) also remains nonnegative. For convenience in calculations we consider the following system, which can be obtained from (2.1) and (2.3): (3.1) D α S = bn pbe qbi r SI N d(n)s, D α E = pbe + qbi + r SI βe d(n)e, N D α I = βe θi γi d(n)i, D α N = N(b d(n)) θi, with initial conditions (3.2) S () = S, E () = E, I () = I, N () = N. 4. Equilibrium points and stability Consider the initial value problem (3.1)-(3.2) with α satisfying < α 1. To evaluate the equilibrium points of (3.1), let D α S =, D α E =, D α I =, D α N =. Then the equilibrium points are F = (K,,, K) and F 1 = (S, E, I, N ), where S = (d(n ) + β pb)(d(n ) + θ + γ) qbβ (4.1) N rβ E = (d(n ) + θ + γ)(b d(n )) (4.2) N βθ I = b d(n ) (4.3) N. θ Here m := d(n ) is the positive root of the following equation: (4.4) (r θ)d 3 (N ) + [(r θ)(β + γ + θ) b(r pθ)]d 2 (N ) rbβγ + [β(r θ)(θ + γ) b(r pθ)(θ + γ) bβ(r qθ)] d(n ) =. Indeed, for r > θ one can show that (4.4) has only one positive root using Descartes rule of signs. The Jacobian matrix J(F ) for the system given by (3.1), evaluated at the disease free equilibrium is as follows: b pb qb r b d (K)K (4.5) J(F ) = (p 1)b β qb + r β b θ γ θ d (K)K 4.1. Theorem. Disease free equilibrium of the system (3.1) is asymptotically stable if β (qb + r) (b + β bp)(γ + θ + b) < 1. Proof. Disease free equilibrium is asymptotically stable if all of the eigenvalues, λ i, i = 1,2, 3,4, of J (F ) satisfy the following conditions [2,11]: (4.6) arg λ i > α π 2.

5 Fractional Order SEIR Model 291 These eigenvalues can be determined by solving the characteristic equation det(j(f ) λi) =. Thus, we have the following algebraic equation: where (λ + b)(λ + kd (k))[λ 2 + (A + B)λ + AB C] = A = b + θ + γ, B = b pb + β, C = β(qb + r). If AB > C, then the condition given by (4.6) is satisfied Remark. Consider the threshold value β (qb + r) R = (b + β bp)(γ + θ + b), which is the basic reproduction number of the system (3.1). The biological interpretation of R is that the disease will take off if R exceeds 1, and will die out if R is less than 1. We now discuss the asymptotic stability of the endemic (positive) equilibrium of the system given by (3.1). The Jacobian matrix J(F 1) evaluated at the endemic equilibrium is given by: where J(F 1) = B 1 = b m θ m rb 1 pb rb 2 B 3 b + (B 2 B 3 B 4 )(rb 1 d (N )N ) rb 1 rb 3 rb 2B 3 rb 1 (B 2B 3 B 4 ) B 1B 2d (N )N β βb 2 B 1 d (N )N θ b m d (N )N, B 2 = m + θ + γ β, B 3 = m + β pb, B 4 = qb r r. So, the characteristic equation of the linearized system is of the form (4.7) λ 4 + a 1λ 3 + a 2λ 2 + a 3λ + a 4 =, with a 1, a 2, a 3 and a 4 being a 1 = b + 2m + rb 1 + rb 3 + βb 2 + d (N )N a 2 = (mr rb + bpr)b 1 + β(2m b)b 2 + r(2m b)b 3 + [m + (r θ)b 1 + rb 3 + βb 2]d (N )N (4.8) + r 2 B 1B 3 bm + m 2 + rβb 1B 2 a 3 = (mrb 3 + bprb 1 + r 2 B 1B 3 + βb 2(m + rb 2) θb1(m rb 1) βθb 1B 2)d (N )N θb 1(prbB 1 + mβb 2 + mrb 3 + r 2 B 1B 3) + r(mβ bβ + bp)b 1B 2 + rθb 1(βB 4 + B 3) + rβ(r θ)b 1B 2B 3 a 4 = brβθb 1 θbβprb1b rβ(mr br mθ)b 1B 2B 3 + mrβθb 1B 4[ bprθb1 2 + β(bpr mθ rθb 1)B 1B 2 + rθ(m rb 1)B 1B 3 + rβθb 1B 4 + rβ(r θ)b 1B 2B 3]d (N )N

6 292 E. Demirci, A. Unal, N. Özalp 4.3. Theorem. Let a 4 be as given in (4.8). If a 4 < then the positive equilibrium point F 1 of the system (3.1) is unstable. Proof. If a 4 <, from Descartes rule of signs it is clear that the characteristic equation (4.7) has at least one positive real root. So, the equilibrium point F 1 of the system (3.1) is unstable. 5. Numerical methods and simulations For the numerical solutions of a system of fractional differential equations, using an Adam s-type predictor corrector method is appropriate [5]. For the parameters (5.1) b =.1555, p =.8, q =.95, r =.5, β =.5, θ =.2, γ =.3, d (N) = N with the initial conditions (5.2) S () = 14, E () =.1, I () =.2, N () = 141, there exists a positive fixed point S = , E = , I = 36.6, N = for IVP (3.1) (3.2). The approximate solutions S(t), E(t), I(t) and N(t) are displayed in Figures 1 4 for α = 1,.95, Concluding Remarks We have formulated and analysed an SEIR model with vertical transmission within a population having a density dependent death rate. We have obtained a stability condition for disease free equilibrium, and a nonstability condition for positive equilibrium. We have also given a numerical example and verified our results. Figure 1. Size of the suspended class over time for system (3.1) with parameters (5.1) and initial conditions (5.2) for different values of α 15 1 alpha 1 alpha.95 alpha.9 Suspended t

7 Fractional Order SEIR Model 293 Figure 2. Size of the exposed class over time for system (3.1) with parameters (5.1) and initial conditions (5.2) for different values of α alpha 1 alpha.95 alpha.9 Exposed t Figure 3. Size of the infectious class over time for system (3.1) with parameters (5.1) and initial conditions (5.2) for different values of α 1 8 alpha 1 alpha.95 alpha.9 Infectious t

8 294 E. Demirci, A. Unal, N. Özalp Figure 4. Size of the total population over time for system (3.1) with parameters (5.1) and initial conditions (5.2) for different values of α 15 1 N 5 alpha 1 alpha.95 alpha t We should note that although the equilibrium points are the same for both integer order and fractional order models, the solution of the fractional order model tends to the fixed point over a longer period of time. We also need to mention that when dealing with real life problems, the order of the system can be determined by using the collected data. References [1] Ahmed, E., El-Sayed, A.M.A. and El-Saka, H. A.A. On some Routh Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems, Physics Letters A 358, 1 4, 26. [2] Ahmed, E., El-Sayed, A.M.A. and El-Saka, H.A.A. Equilibrium points, stability and numerical solutions of fractional-order predator prey and rabies models, JMAA 325, , 27. [3] Brauer, F. and Castillo-Chavez, C. Mathematical Models in Population Biology and Epidemiology (Springer-Verlag, New York, 21). [4] Debnath, L. Recent applications of fractional calculus to science and engineering, IJMMS 54, , 23. [5] Diethelm, K, Ford, N.J. and Freed, A.D. A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics 29, 3 22, 22. [6] Ding, Y. and Ye, H. A Fractional-order differential equation model of HIV infection of CD4 + T-cells, Mathematical and Computer Modeling 5, , 29. [7] El-Sheikh, M. M. A. and El-Marouf, S. A. A. On stability and bifurcation of solutions of an SEIR epidemic model with vertical transmission, IJMMS 56, , 24. [8] Hethcote, H. The mathematics of infectious diseases, SIAM Reviews 43(4), , 2. [9] Kermack, W. O. and McKendrick, A. G. Contributions to the mathematical theory of epidemics, Bulletin of Mathematical Biology 53, 33 55, [1] Lin, W. Global existence theory and chaos control of fractional differential equations, JMAA 332, , 27.

9 Fractional Order SEIR Model 295 [11] Matignon, D. Stability results for fractional differential equations with applications to control processing, in: Computational Eng. in Sys. Appl. 2 (Lille, France, 1996), p 963. [12] McCormack, R. K. and Allen, L. J. S. Disease emergence in multi-host epidemic models, Math. Medicine and Biology 24, 17 34, 27. [13] Ngwa, G. A. and Shu, W.S. A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling 32(7-8), , 2. [14] Odibat, Z. M. and Shawagfeh, N. T. Generalized Taylor s formula, Applied Mathematics and Computation 186, , 27. [15] Podlubny, I. Fractional Differential Equations (Academic Press, London, 1999).